Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
Let $\lambda \in \mathbb{R}$ be a D993: Real number.
(i) | $[a, b] \subset \mathbb{R}$ is a D544: Closed real interval |
(ii) | \begin{equation} a < b \end{equation} |
(iii) | \begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation} |
Then
\begin{equation}
\mathbb{E} \left( e^{\lambda (X - \mathbb{E} X)} \right)
\leq \exp \left( \frac{\lambda^2 (b - a)^2}{8} \right)
\end{equation}